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Parallel and Perpendicular Lines
Parallel Lines
• Two lines with the same slope are said to be parallel lines. If you graph them they will never intersect.
• We can decide algebraically if two lines are parallel by finding the slope of each line and seeing if the slopes are equal to each other.
• We can find the equation of a line parallel to a given line and going through a given point by: a.) first finding the slope m of the given line; b.) finding the equation of the line through the given point with slope m.
Testing if Lines are Parallel
Are the lines parallel? 12 3 9 and -8 2 14x y x y
Find the slope of 12 3 9
3 12 9
4 3
x y
y x
y x
The slope m = -4
Find the slope of 8 2 14
2 8 14
4 7
x y
y x
y x
The slope m = -4
Since the slopes are equal the lines are parallel.
Graphs of Parallel Lines
The red line is the graph of
y = – 4x – 3
and the blue line is the graph of
y = – 4x – 7
Practice Testing if Lines are Parallel
Are the lines 6 3 5 and 2 4 4x y y x parallel? (click mouse for answer)
6 3 5
3 6 5
523
2
x y
y x
y x
m
2 4 4
2 2
2
y x
y x
m
Since the slopes are different
the lines are not parallel.
Are the lines 2 4 and 2 4 12x y x y parallel? (click mouse for answer)
2 4
2 4
1 22
12
x y
y x
y x
m
2 4 12
4 2 12
1 32
12
x y
y x
y x
m
Since the slopes are equal
the lines are parallel.
Constructing Parallel Lines
Find the equation of a line going through the point (3, -5) and
parallel to 2 83
y x
Using the point-slope equation where the slope m = -2/3and
the point is (3, -5) we get 25 33
25 23
2 33
y x
y x
y x
Practice Constructing Parallel Lines
Find the equation of the line going through the point (4,1) and
parallel to (click mouse for answer)3 7y x
1 3 4
1 3 12
3 13
y x
y x
y x
Find the equation of the line going through the point (-2,7) and
parallel to (click mouse for answer)2 8x y
7 2 2
7 2 2
7 2 4
2 3
y x
y x
y x
y x
Perpendicular Lines
• Perpendicular lines are lines that intersect in a right angle.
• We can decide algebraically if two lines are perpendicular by finding the slope of each line and seeing if the slopes are negative reciprocals of each other. This is equivalent to multiplying the two slopes together and seeing if their product is –1.
• We can find the equation of a line perpendicular to a given line and going through a given point by:
a.) first finding the slope m of the given line;
b.) finding the equation of the line through the given point with slope = –1 /m.
Testing if Lines Are Perpendicular
1Are the lines 2 5 and 4 perpendicular?
2x y y x
Find the slope of 2 5 2
2 5
x y m
y x
1 1Find the slope of 4
2 2y x m
Since the slopes are negative reciprocals of each other the lines
are perpendicular. 12 1
2
Graphs of Perpendicular Lines
The red line is the graph of
y = – 2x + 5
and the blue line is the
graph of
y = – 1/2 x +4
Practice Testing if Lines Are Perpendicular
Are the lines 6 3 5 and 2 4 4 perpendicular?x y y x
6 3 5
3 6 5
523
2
x y
y x
y x
m
2 4 4
2 2
2
y x
y x
m
Since the slopes are not
negative reciprocals of
each other (their product
is not -1) the lines are
not perpendicular
Are the lines 2 4 and 4 2 6 perpendicular?x y x y
2 4
2 4
1 22
12
x y
y x
y x
m
4 2 6
2 4 6
2 3
2
x y
y x
y x
m
Since the slopes are
negative reciprocals of
each other (their
product is -1) the lines
are perpendicular.
Constructing Perpendicular Lines
Find the equation of a line going through the point (3, -5) and
perpendicular to 2 83
y x
The slope of the perpendicular line will be m = 3/2 Using
the point-slope equation where the slope m = 3/2 and
the point is (3, -5) we get 35 3
2
3 952 2
3 192 2
y x
y x
y x
Practice Constructing Perpendicular Lines
Find the equation of the line going through the point (4,1) and
perpendicular to (click mouse for answer)3 7y x
11 43
1 413 3
1 13 3
y x
y x
y x
Find the equation of the line going through the point (-2,7) and
perpendicular to (click mouse for answer)2 8x y
17 22
17 22
17 12
1 82
y x
y x
y x
y x
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